On the Complexity of Closest Pair via Polar-Pair of Point-Sets - PowerPoint PPT Presentation

On the Complexity of Closest Pair via Polar-Pair of Point-Sets Bundit Laekhanukit Max-Planck-Institute for Informatics, Germany Joint work with Roee David and Karthik CS

This Talk ● Complexity of Closest Pair ● Geometric Representation of Graphs

Closest Pair (CP) Given a collection of n points in a d-dimensional metric, find a pair of points with minimum-distance.

Known Algorithms ● Euclidean Closest Pair – Dimension d=O(1): ● O(2 D n log n) (deterministic) [Bently-Shamos'76] ● O(2 D n) (randomized) [Rabin'76, Khuller-Mattias'95] – Dimension d=Θ(log n) ● O(d n 2 ) (trivial algorithm) – Dimension d=n: O(n 3-ε ), for some ε > 0

Is there an O(n 1.9 )-time algorithm when dimension d=  (log n)?

Is there an O(n 1.9 )-time algorithm when dimension d=  (log n)? Don't know for Euclidean Closest Pair. No for the bichromatic variant. [Alman-Williams 2015]

Bi-Chromatic Closest Pair (BCP) Given a collection of n red and n blue points in a d-dimensional metric, find a pair of red-blue points with minimum-distance.

? Closest Pair Random Coloring Bi-Chromatic Closest Pair

? Closest Pair If this direction is true, then there is no O(n 1.9 )-time algorithm for Closest Pair. Random Coloring Bi-Chromatic Closest Pair

Closest Pair Exists for L p -metrics for p > 2 via random codes. Random Coloring Bi-Chromatic Closest Pair

Reduction BCP → CP Concatenate point-vectors with codewords Point Vector Red Codeword Point Vector Blue Codeword

Reduction BCP → CP Concatenate point-vectors with codewords Point Vector Red Codeword Point Vector Blue Codeword Needed Properties of The Codewords (Bi-Clique Property) Distance(Red-Code, Red-Code') ≥ R + 1/n Distance(Blue-Code,Blue-Code') ≥ R + 1/n Distance(Red-Code, Blue-Code) = R

The existence of Codewords with Bi-Clique Property implies BCP → CP (that runs in O(n 1.9 )-time)

Complexity Question of CP reduces to Geometric Representation of Bi-Clique R R R

Complexity Question of CP reduces to Geometric Representation of Bi-Clique What is the smallest dimension to represent a bi-clique in L p -metric? (contact-dimension of bi-clique (bicd))

CP & BCP are equivalent in O(log n)-dimension L p -metrics (for p>2)

CP & BCP are equivalent in O(log n)-dimension L p -metrics (for p>2) How about other L p -metrics?

Known Bounds for Bi-Clique Contact Dimension n ≤ bicd(L 0 ) ≤ n ? ≤ bicd(L 1 ) ≤ n 2 ? ≤ bicd(L p ) ≤ n for 1 < p < 2 Maehara 1985 1.286 n ≤ bicd(L 2 ) ≤ 1.5 n Frankl-Maehara 1988 Ω(log n) ≤ bicd(L p ) ≤ O(log n) for p > 2 Ω(log n) ≤ bicd(L ∞ ) ≤ 2 log 2 n

Known Bounds for Bi-Clique Contact Dimension n ≤ bicd(L 0 ) ≤ n ? ≤ bicd(L 1 ) ≤ n 2 ? ≤ bicd(L p ) ≤ n for 1 < p < 2 1.286 n ≤ bicd(L 2 ) ≤ 1.5 n Ω(log n) ≤ bicd(L p ) ≤ O(log n) for p > 2 Ω(log n) ≤ bicd(L ∞ ) ≤ 2 log 2 n David, Karthik CS, L. 2018 Bi-Clique has no contact-graph in O(log n)-dimension for most L p -metrics.

Open Problems ● What is the lower bound for bicd(L 1 ) ? – Related to Kusner's conjecture on equilateral dimension of L 1 – For clique, Alon-Pavel shows contact-dim ≥ Ω(n / log n) ● Better Lower / Upper Bounds for L 1 and L 2 ? ● An alternative way to reduce BCP → CP ? ● One will need a white-box reduction.

The End Thank you for your attention. Questions?

The End Thank you for your attention. Questions? Karthik C.S. will give a 20 min talk in SoCG'18 next week.